First low-frequency Einstein@Home all-sky search for continuous gravitational waves in Advanced LIGO data

We report results of a deep all-sky search for periodic gravitational waves from isolated neutron stars in data from the first Advanced LIGO observing run. This search investigates the low frequency range of Advanced LIGO data, between 20 and 100 Hz, much of which was not explored in initial LIGO. The search was made possible by the computing power provided by the volunteers of the Einstein@Home project. We find no significant signal candidate and set the most stringent upper limits to date on the amplitude of gravitational wave signals from the target population, corresponding to a sensitivity depth of $48.7[1/\sqrt{\mathrm{Hz}}]$. At the frequency of best strain sensitivity, near 100 Hz, we set 90% confidence upper limits of $1.8\times 10^{-25}$. At the low end of our frequency range, 20 Hz, we achieve upper limits of $3.9\times 10^{-24}$. At 55 Hz we can exclude sources with ellipticities greater than $10^{-5}$ within 100 pc of Earth with fiducial value of the principal moment of inertia of ${10}^{38}\mathrm{kg}{\mathrm{m}}^2$.

Figure 1

Number of searched templates in 50 mHz band as a function of frequency. The sky resolution increases with frequency causing the variation in the number of templates. $N_f\times N_{\dot{f}}\sim 1.3\times {10}^9$, where $N_f$ and $N_{\dot{f}}$ are the number of $f$ and $\dot{f}$ templates searched in 50 mHz bands. The total number of templates searched between 20 and 100 Hz is $3\times {10}^{17}$.

Figure 2

On the vertical axis and color-coded is the ${\widehat{\beta }}_{\mathrm{S}/\mathrm{G}\mathrm{Lt}\mathrm{L}}$ in three 50-mHz bands. The top band was marked as “undisturbed.” The middle band is an example of a “mildly disturbed band.” The bottom band is an example of a “disturbed band.”

Figure 3

This is an example of an “undisturbed band” but containing a fake signal. On the $z$ axis and color coded is the ${\widehat{\beta }}_{\mathrm{S}/\mathrm{G}\mathrm{Lt}\mathrm{L}}$.

Figure 4

Highest $2\overline{\mathcal{F}}$ value (also referred to as the $2\overline{\mathcal{F}}$ of the loudest candidate) in every half-Hz band as a function of band frequency. Since the number of templates increases with frequency, so does the highest $2\overline{\mathcal{F}}$. The highest expected $2\overline{\mathcal{F}}\pm 1\sigma (2\sigma )$ over $N_{\text{trials}}$ independent trials is indicated by the darker (faded) band. Two half-mHz bands have $2\overline{\mathcal{F}}$ values greater than the axes boundaries. The half-Hz bands beginning at 33.05 Hz and 35.55 Hz have loudest $2\overline{\mathcal{F}}$ values of 159 and 500, respectively, due to features in the 33.3 Hz and 35.75 Hz 50-mHz bands which were marked “mildly disturbed” in the visual inspection.

Figure 5

Distribution of $p$ values, with binomial uncertainties, for the highest detection statistic values measured in half-Hz bands (circles) and expected from pure Gaussian noise (line). We note that the measured $p$ values for the highest $2\overline{\mathcal{F}}$ in the 33.05 Hz and 35.55 Hz bands are not shown because they are outside of the $x$ axis boundaries.

Figure 6

Candidates that are followed up in stage 1: the distribution of their detection statistic values ${\widehat{\beta }}_{\mathrm{S}/\mathrm{G}\mathrm{Lt}\mathrm{L}}$ (left) and their distribution as a function of frequency (right).

Figure 7

Detection statistic of the loudest candidate from each stage 1 search: the distribution of their detection statistic values ${\widehat{\beta }}_{\mathrm{S}/\mathrm{G}\mathrm{Lt}\mathrm{L}}$ (left) and their distribution as a function of frequency (right). 411 candidates have ${\widehat{\beta }}_{\mathrm{S}/\mathrm{G}\mathrm{Lt}\mathrm{L}}$ values lower than the axes boundaries on the right plot. The red line marks ${\widehat{\beta }}_{\mathrm{S}/\mathrm{G}\mathrm{Lt}\mathrm{L}}=6.0$ which is the threshold at and above which candidates are passed on to stage 2.

Figure 8

Detection statistic of the loudest candidate from each stage 2 search: the distribution of their detection statistic values ${\widehat{\beta }}_{\mathrm{S}/\mathrm{GL}}$ (left) and their distribution as a function of frequency (right). The red line marks ${\widehat{\beta }}_{\mathrm{S}/\mathrm{GL}}=6.0$ which is the threshold at and above which candidates are passed on to stage 3.

Figure 9

Detection statistic of the loudest candidate from each stage 3 search: the distribution of their detection statistic values ${\widehat{\beta }}_{\mathrm{S}/\mathrm{GL}}$ (left) and their distribution as a function of frequency (right). The red line marks ${\widehat{\beta }}_{\mathrm{S}/\mathrm{GL}}=6.0$ which is the threshold below which candidates are discarded.

Figure 10

90% confidence upper limits on the gravitational wave amplitude of continuous gravitational wave signals with frequency in 0.5 Hz bands and with spin-down values within the searched range. The lowest set of points (black circles) are the results of this search. The empty circles denote half-Hz bands where the upper limit value does not hold for all frequencies in that interval. A list of the excluded frequencies is given in the Appendix. The lighter grey region around the upper limit points shows the 11% relative difference bracket between upper limits inferred with the procedure described in Sec. 5 and upper limits that would have been derived (at great computational expense) with direct measurements in all half-Hz bands. We estimate that less than $\sim 0.5\%$ of the upper limit points would fall outside of this bracket if they were derived with the direct-measurement method in Gaussian noise. For comparison we also plot the most recent upper limits results in this frequency range from O1 data obtained with various search pipelines [38]. The better sensitivity of this search is due to the long coherent observation time used. We note also that the searches of [38] cover a broader frequency and spin-down range than the search presented here. All upper limits presented here are population-averaged limits over the full sky and source polarization.

Figure 11

Ellipticity $ϵ$ of a source at a distance $D$ emitting continuous gravitational waves that would have been detected by this search, assuming a source $\dot{f}$ within the range covered by this search. The dashed line shows the spin-down ellipticity for the highest magnitude spin-down parameter value searched, $2.6\times {10}^{-9}\mathrm{Hz}/\mathrm{s}$. The spin-down ellipticity is the ellipticity necessary for all the lost rotational kinetic energy to be emitted in gravitational waves. If we assume that the observed spin-down is all actual spin-down of the object, then no ellipticities could be possible above the dashed curve. In reality the observed and actual spin-down could differ due to radial acceleration of the source. In this case the actual spin-down of the object may even be larger than the apparent one. In this case our search would be sensitive to objects with ellipticities above the dashed line.